ON THE CONSISTENCY OF POSTERIOR MIXTURES AND ITS APPLICATIONS

被引:15
作者
DATTA, S
机构
关键词
POSTERIOR; CONSISTENCY; MIXING DISTRIBUTION; EMPIRICAL BAYES; ASYMPTOTIC OPTIMALITY;
D O I
10.1214/aos/1176347986
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
Consider i.i.d. pairs (theta-i, X-i), i greater-than-or-equal-to l, where theta-1 has an unknown prior distribution omega and given theta-1, X-1 has distribution P theta-1. This setup aries naturally in the empirical Bayes problems. We put a probability (a hyperprior) on the space of all possible omega and consider the posterior mean omega of omega. We show that, under reasonable conditions, P omega = integral-P-theta d omega is consistent in L1. Under a identifiability assumption, this result implies that omega is consistent in probability. As another application of the L1 consistency, we consider a general empirical Bayes problem with compact state space. We prove that the Bayes empirical Bayes rules are asymptotically optimal.
引用
收藏
页码:338 / 353
页数:16
相关论文
共 19 条
[11]  
GILLILAND DC, 1986, IMS LECTURE NOTES MO, V8, P129
[12]  
Hannan J., 1957, CONTRIBUTIONS THEORY, V3, P97
[13]  
HANNAN J, 1960, CPS, P249
[14]   BAAYES ESTIMATION OF MIXING DISTRIBUTION, DISCRETE CASE [J].
MEEDEN, G .
ANNALS OF MATHEMATICAL STATISTICS, 1972, 43 (06) :1993-1999
[15]   SOME ADMISSIBLE EMPIRICAL BAYES PROCEDURES [J].
MEEDEN, G .
ANNALS OF MATHEMATICAL STATISTICS, 1972, 43 (01) :96-&
[16]  
Parthasarathy K. R., 1967, PROBABILITY MEASURES, V3
[17]  
ROBBINS H, 1956, 3RD P BERK S MATH ST, V1, P157
[18]   BAYESIAN ESTIMATION OF MIXING DISTRIBUTIONS [J].
ROLPH, JE .
ANNALS OF MATHEMATICAL STATISTICS, 1968, 39 (04) :1289-&
[19]  
Zarankiewicz K., 1951, COLLOQ MATH-WARSAW, V2, P131